The present invention relates to a method for the evolutionary optimization, to a computer software program for executing such a method as well as to the use of such a method for the optimization of aerodynamic structures.
With reference to FIG. 1 at first the well-known cycle of an evolutionary algorithm will be explained.
In a step 1 the object parameters to be optimized are encoded in a string called individual. A number of such individuals are randomly generated that comprises the initial parent generation and the quality (fitness) of each individual in the parent generation is evaluated. In a step S2 the parents are reproduced by applying genetic operators called mutation and recombination. Thus, a new generation is reproduced in step S3, which is called the offspring generation. The quality of the offspring individuals is evaluated using a fitness function which is the objective of the optimization in step S4. Finally, depending on the calculated quality value, step S5 selects the best offspring individuals (survival of the fittest) which are used as parents for the next generation cycle if the termination condition in step S6 is not satisfied.
Before evaluating the quality of each individual, decoding may be needed depending the encoding scheme used in the evolutionary algorithm. Note that the steps S2, S3, S4, S5 and S6 are cyclically repeated until the condition for termination of the optimization in step S6 is satisfied.
The algorithm of this evolutionary optimization can be expressed by the following pseudo-code:                t:=0        encode and initialize P(0)        decode and evaluate P(0)        do                    recombine P(t)            mutate P(t)            decode P (t)            evaluate P (t)            P(t+1)=select P(t)            encode P (t+1)            t:=t+1                        until terminate        
Such evolutionary algorithms are known to be robust optimizers that are well suitable for discontinuous and multi-modal objective functions. Therefore, evolutionary algorithms have successfully been applied to mechanical and aerodynamic optimization problems, including preliminary turbine design, turbine blade design, multi-disciplinary rotor blade design, multi-disciplinary wing platform design and a military airframe preliminary design.
Details on the evolutionary algorithms can be found in Dasgupta et al. “Evolutionary Algorithms in Engineering Applications”, Springer-Verlag, 1997 and Miettinnen et al., “Evolutionary Algorithms in Engineering and Computer Science”, John Wiley and Sons. 1999.
One essential difficulty in generation-based approaches to aerodynamic optimization is the huge time consumption due to the high complexity of the aerodynamic analysis and the large number of evaluations needed in the evolutionary optimization. To address this problem, several methods have been developed for constructing approximate models.
One widely used method in design engineering is the Response Surface Methodology, which uses low-order polynomials and the least square estimations.
A more statistically sound method is the Kriging model, which is also called the Design and Analysis of Computer Experiments (DACE) model. In this method, a global polynomial approximation is combined with a local Gaussian process and the Maximum Likelihood is used for parameter estimation. Artificial neural networks, including Multi-layer Perceptrons (MLP) and Radial Basis Function Networks (RBFN) have also been employed to build approximate models for design optimization.
A MLP neural network can be used for building an approximate model. However, it is generally difficult to get a model with sufficient approximation accuracy. One of the problems is the lack of training data for the neural network, because for many real application data collection is a computationally expensive process. This is especially true when the dimension of the problem space is high. Due to this, the approximate model may be of low fidelity and may even introduce false optima. In this case, measures need to be taken to guarantee the correct convergence of the optimization algorithm when approximate models are used.
Managing approximate models in optimization based on evolutionary algorithms has not caught much attention so far. It is known to use a heuristic convergence criterion to determine when the approximate model must be updated. The basic idea is that the convergence of the search process should be stable and therefore, the change of the best solution should not be larger than a user-defined value. An assumption is that the first sets of data points are at least weakly correlated with the global optimum of the original problem, which is not necessarily true for high dimensional systems.
An approach to coupling approximate models with evolutionary algorithm has been proposed in an attempt to balance the concern of optimization with that of design of experiments. The main idea is to maintain the diversity of the individuals and to select those data points that are not redundant for model updating (online learning). In this method, the decision of when to carry out the on-line learning of the approximate model is simply based on a prescribed generation delay.
Most recently it has been proposed to train a neural network with some initial samples to approximate the NK model. During evolution, the fittest individual in the current population is evaluated on the original fitness function for every 50 generations. This individual then replaces the one with the lowest fitness in the training set and the neural network is retrained. It has been found that the evolutionary algorithm becomes misled by the neural network model when the complexity of the original fitness landscape is high. The common weakness in the above methods is that neither the convergence properties of the evolutionary algorithm with approximate fitness functions (correct convergence is assumed) nor the issue of model management is addressed.